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Pure
Exchange
Chapter 6
Slides by Pamela L. Hall
Western Washington University
©2005, Southwestern
Introduction


Economists have different treatments for
addressing economic problems
If problem of allocation is generally localized in one
market
 Partial-equilibrium analysis would provide correct solution
• Only one segment of an economy is analyzed


Without consideration for possible interactions with other segments
If it is a general problem infecting numerous
markets or whole economy
 Use general equilibrium analysis
• Study of interaction among agents across markets within an
economy
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Introduction



A (general) equilibrium model of all markets, where supply
and demand for each commodity are equated
 Will result in necessary conditions for economic efficiency
Achieved by agents trading commodities to increase their
utility
 Agents will trade until all gains are exhausted
Efficiency gains from agents’ trading are most apparent
when households are the only agents
 Initially endowed with some quantities of commodities, and there is
no production
• Called a pure-exchange economy

Supply of each commodity is sum of each household’s endowment of that
commodity
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Introduction

This chapter
 Explores gains from trade in a pure exchange economy
• Edgeworth box is a method of illustrating these gains for two
traders
 Investigates efficiency of a free-market price system for
allocating commodities
• Relate this to Pareto-efficient allocation using First and Second
Fundamental Theorems of Welfare Economics
 Develops trial-and-error process of establishing an
equilibrium set of prices
• Relates this to optimal social-welfare allocation
 Discusses a fair allocation of initial resources
• Yielding an optimal social-welfare allocation
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Gains from Trade


In pure-exchange economies, we assume a certain amount of various
commodities exist
Problem is to efficiently allocate these commodities among households
 An allocation of existing commodities is efficient if no one household can be

made better off without making some other household worse off
Necessary condition for such an efficient allocation of commodities is
• MRS1 = MRS2 = … = MRSn




Subscripts denote households
n represents number of households
MRS measures how much a household is willing to trade one commodity for another
When how much each household is willing to trade one commodity for
another are equal
 Gains from trade are exhausted
• Any reallocation of commodities will not increase utility of one household without
decreasing utility of another
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Two-Commodity and TwoHousehold Economy
Consider an economy with two commodities,
bread and fish, and two individuals
(households), Robinson (R) and Friday (F )
 50 units of bread and 100 units of fish are to
be allocated
 Can be allocated in various ways

• Could be all allocated to Robinson, all to Friday, or
•
some combination in between
Egalitarian allocation would divide commodities
equally between Robinson and Friday

Such an allocation may not be efficient if Robinson’s and
Friday’s MRSs are not equal at this equal allocation
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Two-Commodity and TwoHousehold Economy

Suppose Robinson’s MRSR (bread for fish) = 2/1 at an equal allocation
 where Robinson obtains 25 units of bread and 50 units of fish


Friday’s MRSF (bread for fish) = 1/1
Allocation does not result in an efficient allocation because
 MRSR = 2 ≠ 1 = MRSF

If 2 units of bread are taken from Robinson with 1 unit traded to Friday
for 1 fish
 Level of utility remains the same with 1 unit of bread leftover

• Indicated in Table 6.1
By trading, utilities of Robinson and Friday remain unchanged, with 1
unit of bread left over
 Represents gains from trade
• Could then be divided between Robinson and Friday

Resulting in their utility increasing
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Table 6.1 Gains from trade
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Two-Commodity and TwoHousehold Economy
As long as MRSs between Robinson and
Friday are unequal
 Gains from trade are possible
 Many possible trades that will result in gains
 Use Robinson’s and Friday’s initial
endowments and indifference curves to
determine all possible trades leading to gains
 Shown in Figure 6.1

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Figure 6.1 Preferences and endowments
for the two-commodity…
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Edgeworth Box


Provides a convenient method for representing the two
households’ preferences and endowments in one diagram
 See Figure 6.2
Construct box by turning preference space for Friday 180
 Place it on top of Robinson’s preference space at point where their
endowments are together
• Point C in Figure 6.2


Horizontal width represents total quantity of fish available
Vertical height represents total quantity of bread available
 Size of box depends on total amount of fish and bread available in
economy
 Every point inside box represents a feasible allocation of fish and
bread
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Figure 6.2 Edgeworth box in a
pure-exchange economy
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Edgeworth Box

An allocation is feasible if total quantity consumed of each commodity is equal to
total available from endowments
 xR1 + xF1 = eR1 +eF1, commodity 1, fish
 xR2 + xF2 = eR2 +eF2, commodity 2, bread

An allocation where Robinson receives nothing and Friday receives all is a
feasible allocation
 Represented by 0R in Figure 6.2
• Robinson’s utility is minimized and Friday’s utility is maximized

At 0F, allocation is nothing for Friday and everything for Robinson
 Friday’s utility is minimized and Robinson’s maximized

Feasible allocations between these two extreme points represent combinations
of commodities with varying levels of satisfaction for both Robinson and Friday
 For a movement toward 0R, Friday receives more of either fish or bread
• Increases her utility
 Robinson receives less, which decreases his utility
• Reverse occurs for a movement toward 0F
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Pareto-Efficient Allocation


Allocation where there is no way to make all
households better off
No way to make some households better off
without making someone else worse off
 All gains from trade are exhausted
• Illustrated in Figure 6.3
• Point C is not Pareto efficient

Possible to reallocate commodities in such a manner that one
household can be made better off without making another worse off
• Any point within shaded lens represents a gain (Pareto
improvement)
• At points A, B, and all points on the cord

MRSR = MRSF and are Pareto-efficient allocations
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Figure 6.3 Efficiency in a pureexchange economy
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Pareto-Efficient Allocation
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
Cord from points A to B is called the core solution
 All gains from trade are exhausted
Exact solution point within this core depends on
bargaining strength of two agents
 If Friday is a relatively strong bargainer
• She will receive more of gains from trade

Solution will be closer to point B
 Alternatively, if Robinson has the upper hand
• He will receive a larger portion of gains

Solution will approach point A
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Contract Curve (Pareto-Efficient
Allocation)

Optimal allocation resulting from allocating
commodities between Robinson and Friday
 Will depend on how initial total endowment of
bread and fish is divided between them
• If Robinson initially has most of the bread and fish

Optimal allocation near point D in Figure 6.4 may result
 The more fish and bread a household initially
has, the higher the level of utility it can achieve
• Distribution of income determines resulting Paretoefficient allocation
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Contract Curve (Pareto-Efficient
Allocation)

By varying allocation of initial endowments
 Can trace out complete set of Pareto-efficient allocations
• Called a contract curve

Illustrated in Figure 6.4
• Contract curve represents a curve in interior of Edgeworth box

Intersecting tangencies between indifference curves for two agents
 MRSs are equal
• If efficient allocations exist, where an agent will not consume a
positive amount of all commodities

Contract curve will correspond with a segment of an axis (corner
solution)
 MRSs will not equate
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Figure 6.4 Contract curve in a
pure-exchange economy
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Contract Curve (Pareto-Efficient
Allocation)


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Any point not on contract curve is inefficient
Agents will adjust terms of trade until a contract is made
Represents all Pareto-efficient allocations for a given set of
initial endowments
 Every point on contract curve results in economic efficiency
• Social welfare is not maximized at every point


Movement along a contract curve will increase one agent’s
utility at expense of reduced utility for other agent
Maximum social welfare depends on
 Economic efficiency
 Optimal distribution of income
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Contract Curve (Pareto-Efficient
Allocation)

Pareto optimality does provide a necessary condition for an
allocation to maximize social welfare
 No inefficiencies in resource allocation exist
• Necessary condition for maximum social welfare

Major inadequacy of Pareto-welfare criterion
 Does not lead to a complete social ranking of alternative allocations
for an economy
• Useless criterion for many policy propositions

Some analytical results can be obtained with a Paretowelfare criterion
 For example, point rationing is ordinarily better than fixed-ration
quantities
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Efficiency of a Price System
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A medium of exchange is particularly useful as
number of households increases
 Process of bartering becomes cumbersome
All commodities are valued by this medium
 Medium is accepted in exchange for commodities
 Medium of exchange is money
Money is accepted not for direct utility it provides
 But for indirect utility via commodities it can purchase
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Efficiency of a Price System

Allocation device that has received by far the greatest
attention by economists is price system
 Assumes all commodities are valued in market by their money
equivalence

Permits decentralization of allocation decisions
 Provides a method for relating household preferences with supply at
a reduced cost for society
• Prices act as signals to economic agents in guiding their supply and
demand decisions
 Under a perfectly competitive price system, households have no
control over market prices
• Take prices as given

Yields a Pareto-efficient market system
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Efficiency of a Price System

In a perfectly competitive price system
 Correspondence between Pareto-efficient allocation of resources
and perfectly competitive price system is exact
• Called First Fundamental Theorem of Welfare Economics


Provides formal and general confirmation of Adam Smith’s invisible hand
Second Fundamental Theorem of Welfare Economics states
 Every Pareto-efficient allocation has an associated perfectly
competitive set of prices
 States possibility of achieving any desired Pareto-efficient allocation
as a market-based equilibrium using an appropriate distribution of
income
• Not every Pareto-efficient allocation is a social-welfare optimum
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Efficiency of a Price System

Recall that Pareto efficiency in exchange requires
 MRS1 = MRS2 = … = MRSn
• For n households in economy
 For utility maximization subject to a wealth or income constraint,
each household equates its MRS with price ratio
• MRS(x2 for x1) = p1/p2
 Every household faces same price ratio
 Market in equilibrium creates a societal trade-off rate that is a correct
reflection of every household’s trade-off rate
• Information on this trade-off rate (if it could be gathered) would require
large expenditures by a government

Instead, this trade-off can be generated by perfectly competitive interaction
of supply and demand
 At zero governmental cost
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Offer Curve





Traces out points where household maximizes utility for a
given level of income across various ratios of prices
 See Figure 6.5
At each point household’s indifference curve is tangent to a
budget constraint for a given price ratio
Represents how much a household is willing to offer one
commodity in exchange for the other at a given price ratio
Analogous to price consumption curve with focal point at
initial endowment e
At alternative price ratios, endowment is affordable
 Every point on offer curve is at least as good as agent’s endowment
point e
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Figure 6.5 Offer curve
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Offer Curve

Represents a set of demanded bundles
 Each demanded bundle associated with a price
ratio
• As price ratio continues to increase, new demanded
bundles unfold
• Locus of all these demanded bundles is offer curve
• Each household has an offer curve and initial
endowment of commodities
• Relating offer curves and endowment of two
households in an Edgeworth box

Walrasian equilibrium is illustrated in Figure 6.6
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Figure 6.6 Pareto efficiency and
competitive pricing
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Offer Curve

Where two offer curves intersect, point A,
price ratio is same for both Robinson and
Friday
 Demanded bundles exactly match supply
 MRSs for Robinson and Friday are equated
• Yields one-to-one correspondence between Pareto
efficiency and perfectly competitive markets
MRSs are both equal to p1*/p2*
•
 Aggregate supply equals aggregate demand for
each of the commodities
• Households are maximizing their utility
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Walras’s Law



Static equilibrium conditions are generally
theoretical results in economics
Economy does not operate on a set of natural laws
that describe its evolution
Process of how an economy in disequilibrium
reaches an equilibrium state (called tâtonnement
stability) can be described only in limited detail
 First described by Walras
 Tâtonnement is French
• Means groping or trial and error
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Walras’s Law

For example, if at p1'/p2' quantity demanded for fish
is greater than quantity supplied and quantity
demanded for bread is less than quantity supplied
 Price ratio would rise
• Quantity demanded for fish would decline and that for bread
would increase


Adjustment would continue until prices converge to competitive
equilibrium levels
Adjustment assumes prices will respond to market
shortages and surpluses
 If prices are rigid then tâtonnement process will not work
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Walras’s Law



Friday’s demand functions for commodities x1 and
x2, respectively
 x1F(p1, p2, e1F, e2F) and x2F( p1, p2, e1F, e2F)
Robinson’s demand functions
 x1R(p1,p2,e1R,e2R) and x2R( p1, p2,e1R,e2R)
Walrasian equilibrium set of prices (p1*, p2*) is
where aggregate demand equals aggregate supply
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Walras’s Law

Alternatively, this Walrasian equilibrium may be represented
in terms of aggregate excess demand functions
 For zj > 0 commodity j is in excess demand
 For zj < 0 commodity j is in excess supply
• For j = 1, 2
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Walras’s Law

Walrasian equilibrium exists when these
aggregate excess demand functions are zero
 Result of specifying markets for commodities in
terms of excess demand is Walras’s Law
• Value of aggregate excess demand is zero for not only the
equilibrium set of prices (p1*, p2*) but for all possible prices

In two-commodity economy, if z1 > 0 (excess
demand), then, given positive prices, z2 < 0 (excess
supply) for Walras’s Law to hold
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Walras’s Law

Proof of Walras’s Law
 Involves adding up households’ budget constraints and rearranging
terms

Consider Friday’s budget constraint, where the right-hand
side is Friday’s income, represented as value of Friday’s
endowments of x1 and x2
• Total expenditures on x1 and x2 will equal this value of endowments
 Rearranging terms, we get
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Walras’s Law

Define Friday’s excess demands for commodities 1
and 2 as
 For z1F > 0 Friday has excess demand for fish and z1F < 0
Friday has excess supply

Represent Friday’s budget constraint as
 Value of Friday’s excess demand for the two
commodities is zero
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Walras’s Law


Robinson’s value of excess demand for the two
commodities is also zero
Adding Friday’s and Robinson’s value of excess
demand functions yields Walras’s Law
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Relative Prices

Important result of Walras’s Law
 If aggregate demand equals aggregate supply in
one market, then, for a two-market economy
• Demand must equal supply in the other market
 Generalizing to k commodities, if aggregate
demand equals aggregate supply in (k - 1)
markets
• Demand must equal supply in remaining excluded
market
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Relative Prices

By Walras’s Law, if p2* > 0, then
 A set of prices where aggregate demand for one market equals aggregate
supply will result in other market clearing

Mathematically, this implies (k - 1) independent equations in a kcommodity model
 With one less equation than k number of market clearing prices, cannot
solve system for a set of k independent prices
• Can determine only relative prices
 Specifically, in general equilibrium, each household’s income is the value of
endowment at given prices
• Each household’s budget constraint is homogeneous of degree zero in prices
• In general equilibrium, only relative prices are determined, given that all
•
households’ budget constraints are homogeneous of degree zero in prices
Multiplying all prices by some positive constant does not change households’
demand and supply for commodities
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Social Welfare

Walrasian equilibrium may not be an optimal social-welfare point
 However, does result in a Pareto-efficient allocation
 Assumes a given distribution of initial endowments

If endowments are distributed in such a way that one agent receives a
relatively small share of initial endowments and other agent receives a
relatively large share
 Social welfare may not be maximized
• To accomplish optimal distribution of initial endowments along with competitive
prices is required

One criticism of perfect competition (capitalist markets in general)
involves
 Restriction that perfectly competitive markets take this distribution of initial

endowments as given
With any initial distribution of endowments that society deems as “unfair”
• Perfect competition will not maximize social welfare
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Equitable Distribution of
Endowments and Fair Allocations

Problem of determining an optimal distribution of initial
endowments is normative in nature
 Involves value judgments concerning satisfaction households
receive from their endowments
• Possible normative solution is an optimal distribution of initial
endowments that is equitable


May be defined as an allocation of endowments where no household prefers
any other household’s initial endowment
One equitable allocation of initial endowments is an equal
division of commodities
 Each household has same initial commodity bundle
• Equal division will probably not be Pareto efficient

A competitive market, given this initial equal division of commodities, will
yield a Walrasian equilibrium that is Pareto efficient
 Such a market allocation is called a fair allocation
 Both equitable and Pareto efficient
42
Equitable Distribution of
Endowments and Fair Allocations

Can show a fair allocation resulting from a
competitive reallocation of equitable initial
endowments by contradiction
 Assume allocation is not fair and Robinson is envious
• Prefers Friday’s allocation to his own
• Robinson cannot afford Friday’s allocation
• However, equal distribution of initial endowments implies value of
initial endowment must be the same
• Thus, Friday also cannot afford her optimal allocation

Results in a contradiction
43
Equitable Distribution of
Endowments and Fair Allocations

Impossible for Robinson to envy Friday at Pareto-efficient allocation
 So a competitive equilibrium from equitable initial endowments is a fair
allocation

Some societies have achieved this equal division in value of initial
endowments by reducing size of Edgeworth box
 Through war, pestilence, and famine, current and future generations of
households within these societies are or will be at a near subsistence level
• Social welfare is not maximized by achieving an equitable distribution of
endowments

One problem with achieving an equal distribution of initial endowment
values is
 Incentives to work and invest are reduced

• No incentive to provide next generation with additional endowments
In a communist society, working for the common good is meant to
replace these individual incentives
 Results in an enlargement of Edgeworth box for all comrades
• This degree of altruism may be too much to ask of individual households
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Equitable Distribution of
Endowments and Fair Allocations

An alternative to an equal distribution of value of
endowments
 Providing equal opportunities for enriching a household’s
endowments


Equal opportunity was one of driving forces for
large migration of households to United States in
19th century
Providing an initial endowment consisting of equal
opportunities is an equitable allocation
 Provides an underlying justification for equal opportunity
legislation
• Ranging from minority rights to funding for public education
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Equitable Distribution of
Endowments and Fair Allocations

U.S. history indicates that combining equal
opportunity with free markets can greatly enlarge
Edgeworth box
 Results in increasing all households’ utilities and in fair
allocations leading to an optimal social-welfare allocation
of commodities
• However, a great deal of poverty still exists within United States
• The reversal in 1980s of a more equal distribution of wealth
indicates that United States has not reached local bliss
(maximum social welfare)
• Other societies are generally more socialistic than United States

Unwilling to have such a large inequality in wealth
 Generally strive for a more equal distribution of endowments
• Which system is preferred is a value judgment
46